Rectilinear motion problems deal with objects in motion and their position, velocity and acceleration. The position function is treated as the original function and is noted by s(t). The velocity function is the derivative of the position function and is noted by s'(t) or v(t). The acceleration function is the derivative of the velocity function and is noted by s"(t), v'(t) or a(t). Many times the question is set up in this format: A billiard ball is hit and travels in a line. If s centimeters is the distance of the ball from its initial position at t seconds, then s=100t^2 + 100t. So, to figure out how far the ball has traveled from the intial position, one must plug in how long it has been in t seconds. To find out when the ball is going in the positive and negative direction, one must take the derivative of the position function, s(t). When plugging in t to the velocity function, the ball goes in the positive direction when v(t) equals a positive number and same goes for negative direction with a negative velocity. To find out if the ball is speeding up, one must take into account both the velocity function and the acceleration function. If the ball has the same sign for both velocity and acceleration, then the ball or object is speeding up. If the signs are opposite, then the ball or object would be slowing down.
Watch this for further explanation:
https://www.youtube.com/watch?v=Q6MT8uVSx38